We have a Lennox heat pump with a standard (non-programmable) Honeywell thermostat. Last weekend we didn't have any heat, no fan, nothing. My cousin who does commercial heating/air conditioning install & repair replaced the old programmable thermostat with the a new, standard Honeywell thermostat. He also replaced the relay and I believe he said it was the transmitter (I could be wrong) on the internal unit. He was able to get everything running. The problem... now the heat pump doesn't shut off once it hits the desired temperature on the thermostat. We have it set on heat at 68 degrees. The night he replaced the parts it ran for almost 3 hours straight! It was quite chilly in the house (around 61 degrees) because we hadn't had any heat for a few days. So, we expected it. However, once the thermostat hit 68 it kept running. It was almost 77 degrees when we realized it was getting caribbean-like in the house! We had to manually by turning the thermostat to off. We let the house cool down to 70 before turning it back on. We figured it wouldn't come on because we had it set at 68. Wrong! It came on and kept running again. We turned it off when it hit 74. The good news is we have discovered how well insulated our house is! It took 12 hours (through the night with 38 degrees outside) for the house to cool from 77 to 70! The bad news is my cousin came back out and couldn't figure out the problem! Before I call someone for a "free estimate" and taking time off work to get the estimates, anyone have any suggestions? Please!!

You turned it off at the sat and it goes off? Or you have to kill the main power at the breaker box? If you are just turning it off at the stat and it goes off then the stat must be bad, or the white wire in the stat is touching something it is not supposed to be.

We can turn it off at the stat. It's a new stat just bought/installed Wednesday. I'll check the white wire. I know very little about heating/air conditioning repair, so what should the white wire be touching, if anything? Might be a bad stat.

the indoor, and the outdoor unit both keep running??

Yes.